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comp.nus.sg/~cs2100/

http://www.comp.nus.edu.sg/~cs2100/. Lecture #13. Boolean Algebra. Lecture #13: Boolean Algebra. Digital Circuits Boolean Algebra Truth Table Precedence of Operators Laws of Boolean Algebra Duality Theorems Boolean Functions Complement Functions Standard Forms Minterms and Maxterms

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comp.nus.sg/~cs2100/

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  1. http://www.comp.nus.edu.sg/~cs2100/ Lecture #13 Boolean Algebra

  2. Lecture #13: Boolean Algebra Lecture #13: Boolean Algebra • Digital Circuits • Boolean Algebra • Truth Table • Precedence of Operators • Laws of Boolean Algebra • Duality • Theorems • Boolean Functions • Complement Functions • Standard Forms • Minterms and Maxterms • Canonical Forms: Sum-of-Minterms and Product-of-Maxterms

  3. High Low Signals in digital circuit Signals in analog circuit Lecture #13: Boolean Algebra 1. Digital Circuits (1/2) • Two voltage levels • High/true/1/asserted • Low/false/0/deasserted • Advantages of digital circuits over analog circuits • More reliable (simpler circuits, less noise-prone ) • Specified accuracy (determinable) • Abstraction can be applied using simple mathematical model – Boolean Algebra • Ease design, analysis and simplification of digital circuit – Digital Logic Design

  4. Lecture #13: Boolean Algebra 1. Digital Circuits (2/2) • Combinational: no memory, output depends solely on the input • Gates • Decoders, multiplexers • Adders, multipliers • Sequential: with memory, output depends on both input and current state • Counters, registers • Memories

  5. A B A B AB A A+B A' Lecture #13: Boolean Algebra 2. Boolean Algebra • Truth tables • Logic gates • Boolean values: • True (T or 1) • False (F or 0) • Connectives • Conjunction (AND) • A B; A  B • Disjunction (OR) • A + B; A  B • Negation (NOT) • ; ; A' In CS2100, we use the symbols ∙ for AND, + for OR, and ' for negation. Please follow.

  6. Lecture #13: Boolean Algebra 2. Boolean Algebra: AND • Do write the AND operator ∙ (instead of omitting it) • Example: Write a∙b instead of ab • Why? Writing ab could mean that it is a 2-bit value.

  7. Lecture #13: Boolean Algebra 3. Truth Table • Provide a listing of every possible combination of inputs and its corresponding outputs. • Inputs are usually listed in binary sequence. • Example • Truth table with 3 inputs x, y, z and 2 outputs (y + z) and (x  (y + z))

  8. Lecture #13: Boolean Algebra 3. Proof using Truth Table • Prove: x  (y + z) = (x  y) + (x  z) • Construct truth table for LHS and RHS • Check that column for LHS = column for RHS • DLD page 59 Quick Review QuestionsQuestion 3-1. 

  9. Lecture #13: Boolean Algebra 4. Precedence of Operators • Precedence from highest to lowest • Not (') • And () • Or (+) • Examples: • A  B + C = (A  B) + C • X + Y' = X + (Y') • P + Q'  R = P + ((Q')  R) • Use parenthesis to overwrite precedence. Examples: • A  (B + C) [ Without parenthesis: A  B + C ] • (P + Q)'  R [ Without parenthesis: P + Q'  R ]

  10. Lecture #13: Boolean Algebra 5. Laws of Boolean Algebra

  11. Lecture #13: Boolean Algebra 6. Duality • If the AND/OR operators and identity elements 0/1 in a Boolean equation are interchanged, it remains valid • Example: • The dual equation of a+(bc)=(a+b)(a+c) is a(b+c)=(ab)+(ac) • Duality gives free theorems – “two for the price of one”. You prove one theorem and the other comes for free! • Examples: • If (x+y+z)' = x'y'z' is valid, then its dual is also valid:(xyz)' = x'+y'+z' • If x+1 = 1 is valid, then its dual is also valid:x0 = 0

  12. Lecture #13: Boolean Algebra 7. Theorems

  13. Lecture #13: Boolean Algebra 7. Proving a Theorem • Theorems can be proved using truth table, or by algebraic manipulation using other theorems/laws. • Example: Prove absorption theorem X + XY = X X + XY = X1 + XY (by identity) = X(1+Y) (by distributivity) = X(Y+1) (by commutativity) = X1 (by one element) = X (by identity) • By duality, we can also cite (without proof) thatX(X+Y) = X

  14. Lecture #13: Boolean Algebra 8. Boolean Functions • Examples of Boolean functions (logic equations): F1(x,y,z) = xyz' F2(x,y,z) = x + y'z F3(x,y,z) = x'y'z + x'yz + xy' F4(x,y,z) = xy' + x'z From the truth table, F3 = F4. Can you prove F3 = F4 by using Boolean Algebra? 

  15. Lecture #13: Boolean Algebra 9. Complement Functions • Given a Boolean function F, the complement of F, denoted as F', is obtained by interchanging 1 with 0 in the function’s output values. • Example: F1 = xyz' • What is F1' ? • F1' = (xyz')' = x' + y' + (z')' (DeMorgan’s) = x' + y' + z (Involution) 

  16. Lecture #13: Boolean Algebra 10. Standard Forms (1/2) • Certain types of Boolean expressions lead to circuits that are desirable from an implementation viewpoint. • Two standard forms: • Sum-of-Products (SOP) • Product-of-Sums (POS) • Literals • A Boolean variable on its own or in its complemented form • Examples: (1) x, (2) x', (3) y, (4) y' • Product term • A single literal or a logical product (AND) of several literals • Examples: (1) x, (2) xyz', (3) A'B, (4) AB, (5) dg'vw

  17. Lecture #13: Boolean Algebra 10. Standard Forms (2/2) • Sum term • A single literal or a logical sum (OR) of several literals • Examples: (1) x, (2) x+y+z', (3) A'+B, (4) A+B, (5) c+d+h'+j • Sum-of-Products (SOP) expression • A product term or a logical sum (OR) of several product terms • Examples: (1) x, (2) x + yz', (3) xy' + x'yz, (4) AB + A'B', (5) A + B'C + AC' + CD • Product-of-Sums (POS) expression • A sum term or a logical product (AND) of several sum terms • Examples: (1) x, (2) x(y+z'), (3) (x+y')(x'+y+z), (4) (A+B)(A'+B'), (5) (A+B+C)D'(B'+D+E') • Every Boolean expression can be expressed in SOP or POS form. • DLD page 59 Quick Review QuestionsQuestions 3-2 to 3-5.

  18. Lecture #13: Boolean Algebra Quiz Time! SOP expr: A product term or a logical sum (OR) of several product terms. POSexpr: A sum term or a logical product (AND) of several sum terms. • Put the right ticks in the following table.             

  19. Lecture #13: Boolean Algebra 11. Minterms and Maxterms (1/2) • A minterm of n variables is a product term that contains n literals from all the variables. • Example: On 2 variables x and y, the minterms are: x'∙y', x'∙y, x∙y' and x∙y • A maxterm of n variables is a sum term that contains n literals from all the variables. • Example: On 2 variables x and y, the maxterms are: x'+y', x'+y, x+y' and x+y • In general, with n variables we have up to 2nminterms and 2n maxterms.

  20. Lecture #13: Boolean Algebra 11. Minterms and Maxterms (2/2) • The minterms and maxterms on 2 variables are denoted by m0 to m3 and M0 to M3 respectively. • Each minterm is the complement of the corresponding maxterm • Example: m2 = x∙y' m2' = ( x∙y' )' = x' + ( y' )' = x' + y = M2

  21. Lecture #13: Boolean Algebra 12. Canonical Forms • Canonical/normal form: a unique form of representation. • Sum-of-minterms = Canonical sum-of-products • Product-of-maxterms = Canonical product-of-sums

  22. F1 = x∙y∙z' = m6 F2 = x'∙y'∙z + x∙y'∙z' + x∙y'∙z + x∙y∙z' + x∙y∙z = m1 + m4 + m5 + m6 + m7 = Sm(1,4,5,6,7) orSm(1,4 – 7) Lecture #13: Boolean Algebra 12.1 Sum-of-Minterms • Given a truth table, example: • Obtain sum-of-minterms expression by gathering the minterms of the function (where output is 1). F3 = x'∙y'∙z + x'∙y∙z + x∙y'∙z' + x∙y'∙z = m1 + m3 + m4 + m5 = Sm(1,3,4,5) orSm(1,3 – 5) 

  23. F2 = (x+y+z) ∙(x+y'+z) ∙ (x+y'+z') = M0 ∙ M2 ∙ M3 = PM(0,2,3) Lecture #13: Boolean Algebra 12.2 Product-of-Maxterms • Given a truth table, example: • Obtain product-of-maxterms expression by gathering the maxterms of the function (where output is 0). F3 = (x+y+z) ∙ (x+y'+z) ∙ (x'+y'+z) ∙ (x'+y'+z') = M0 ∙ M2 ∙ M6 ∙ M7 = PM(0,2,6,7) 

  24. Lecture #13: Boolean Algebra 12.3 Conversion of Standard Forms • We can convert between sum-of-minterms and product-of-maxterms easily • Example: F2 = Sm(1,4,5,6,7) = PM(0,2,3) • Why? See F2' in truth table. • F2' = m0 + m2 + m3Therefore,F2 = (m0 + m2 + m3)' = m0' ∙ m2' ∙ m3' (by DeMorgan’s) = M0 ∙ M2 ∙ M3 (as mx' = Mx) • Read up DLD section 3.4, pg 57 – 58. • Quick Review Questions: pg 60 – 61, Q3-6 to 3-13.

  25. Lecture #13: Boolean Algebra End of File

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